Mathematics and Reality

Keith Ward on Materialism, 10     1  2  3  4  5  6  7  8  9

Whatever all this means, it has left old-fashioned classical materialism far behind.

It is important to keep in mind that the whole of the argument this far is that science no longer believes in such materialism and that it “has become increasingly hard to say just what ‘matter’ is”.

The ultimate reality is beyond space-time as we know it,

This, on the other hand, is a conclusion from the scientific theories cited that, as far as it goes, not only coincides entirely with idealism but represents a positive formulation of idealism as opposite to materialism.

Ward does not discuss the meaning of the addition “as we know it”, which implies the possibility of space-time as we do not know it, space-time of a kind other than what we know or experience as such.

has a deep and complex mathematical structure, and is nothing like the world we see and touch and feel.

Idealism as I would defend it says more than this. By the world we see and touch and feel, Ward here, I think, means primarily what I called the world of common-sense materialism, but which he is disinclined to designate in this way since he wants to retain a common-sense position and defend the common-sense tradition with regard to the question of God and belief in God. Ultimate reality is certainly nothing like that. But neither is it the mere hypostasized abstraction that is the deep and complex mathematical structure. A mathematical structure is certainly part of it, or an aspect of it, as understood by idealism, but if it were only that, it is not obvious that its reality would be ultimate, that it would be more real than the world we see and touch and feel in terms of concrete experience and content of consciousness. Nor could the mere mathematical structure as ultimate reality account for the appearance or phenomenon of that world. But again, although he states the conclusion in terms that overlap with those of philosophical idealism, Ward is here discussing only the aspect of ultimate reality that seems to be affirmed by contemporary physics.

It is certainly not made of matter, in the sense of solid bits of stuff, precisely located in three dimensional space. Questions like, “Where are the fundamental laws of nature located?”, or “How much time do quantum fluctuations in a vacuum take?” will be met with pitying looks by mathematical physicists. They (the laws and fluctuations, not the physicists) are not anywhere in our space, or at any point in what we ordinarily think of as time.

Ward again uses qualifications like “our” space and “what we ordinarily think of “ as time. As we shall see, he does not really develop their implications.

This meas that the simple-minded materialism that insists that everything that exists must be somewhere, or that everything that exists must exist at some time, is just woefully ignorant of modern physics. There are supra-spatial and supra-temporal realities, realities beyond any and all spaces and times, and mathematical physics talks about them with an immense degree of sophistication and precision.

This paragraph could seem to contradict the preceding qualifications, inasmuch as the supra-spatial and supra-temporal relities are not just beyond space-time “as we know it”, “our” space and “what we ordinarily think of” as time, but “beyond any and all spaces and times”. If the latter formulation is true, possible other spaces and times cannot be the ultimate reality or “parts” or aspects of it.

But if we distinguish between the first formulation above about ultimate “reality” and this formulation about supra-spatial and supra-temporal “realities”, it is possible that ultimate reality could be conceived by Ward as including both the realities that are beyond also the space and time that is other than space and time as experienced by us, and that possible other space-time. If the supra-spatial and supra-temporal realities spoken of here are, in such an ultimate reality, considered superior to or “more ultimate” than the space-time that is not as we know it, it is of course not an ultimate reality; the other space-time aspect of it is not really an ultimate reality.

But again, this view requires that the mere mathematical structure can as such account for that as well as this (our) space-time as such. And if space-time as such, ours as well as other possible ones, cannot be apprehended at all, but is always apprehended along with the concrete experience which in turn requires or is defined in terms of content of consciousness, it also requires that the mathematical structure can account for all of this content too.

Now, it does seem some mathematical physicists do indeed think it is able to do that, and this is certainly what Ward has in mind here. But at least until some mathematician has explained – and here, I think, we find a clear instance of the real inevitability of philosophy for science – the nature of mathematics as being in principle and essence something quite different from what it has heretofore been considered to be, this seems to me philosophically untenable. It presupposes a new conception of mathematics which must be conceived in philosophical terms.

Although certainly aspects of ultimate reality, the supra-temporal and supra-spatial realities of the mathematical model Ward speaks of must be co-ultimate with other aspects which must perforce include that which accounts for phenomenal reality as experienced by us, and may or may not include alternate time-space. Since ultimate reality must account for all of reality, all degrees and levels of reality, it seems obvious that space-time as we know it as well as possible other space-time must somehow be accommodated in the ultimacy of the reality which, in the case of the former space-time, is experientially ascertainable as phenomenal appearance even by us as finite beings.

But there is also another problem here which I briefly pointed to in the last post in this series. When Ward refered to the findings regarding sub-atomic particles, waves, energy, probability-waves, virtual particles, a vacuum (lowest-energy) state, and quantum foam, he was not speaking exclusively of a deep and complex mathematical structure, as such. He was speaking of other entities described in mathematical terms but not in themselves exhaustively reducible to and identical with mathematics. The point was well taken that these entities did not correspond to matter as conceived by “old-fashioned materialists”, and this certainly confirmed the one basic claim he has this far made, namely that it has become increasingly difficult to say what “matter” is.

And above, he said that ultimate reality has a deep and complex mathematical structure. But when he spoke of reformulating the theory of forms as a theory of objective mathematical axioms, he did not just assert that ultimate reality has but that it is such a structure. But since the reference here is not to Plato’s mystic concption of real mathematical entities, but to the reformulation of the theory of forms in terms of the mathematical models of contemporary physics, it could, I think, be asked if a hypostasization is not involved which transforms what is in reality a mere pragmatic conception into something equivalent to the truly absolute conception of mathematics that we find in Plato.

I have argued that the conception of the reducibility of ultimate reality to mathematics is problematic in general (and, I think, including mathematics as conceived by the later Plato), but the problem seems to be compounded when mathematics is conceived in the manner of contemporary physics, i.e. as defining a thoroughgoingly pragmatic science. Is this really what the majority of mathematical physicists claim? Don’t they rather conceive of their mathematics in terms similar to those described by Evola, “a net that draws ever tighter around a something that, in itself, remains incomprehensible, with the sole intention of subduing it for practical ends” – something, whatever it is, at least not being mathematics itself?

In the light of these considerations, it may seem that “matter” is just a sort of thin and abstract skeleton, a desiccated substructure, of the richly observed world of human perceptions.

The “richly observed world of human perceptions” is what I want to regard here as having its counterpart in ultimate reality, in accordance with Plotinus’ conception of Platonic logic in contradistinction to the Aristotelian, a conception which I have mentioned briefly in another post, and with the version of idealism in general that I have discussed at length in other publications.

Although this richly observed world is phenomenal and appearance, it does not mean it is unreal. And although the degree of its reality is lower than that of ultimate reality, the reality is such that it must be matched by something in ultimate reality of which it is a phenomenon and appearance, and that this something cannot be a mathematical model only as such. The mere mathematical model of the kind that is discussed here by Ward as the opposite of matter as conceived by  “old-fashioned classical materialism”, could be seen to be described here as that which has replaced this conception of matter, namely “a sort of thin and abstract skeleton, a desiccated substructure”. And such a substructure cannot be rationally conceived as having by itself in any way produced “the richly observed world of human perceptions”. The latter, the apperarance, would be more than that of which it is the appearance.

But it seems the reason why “matter”, or the appearance of matter, has thus been reduced to a thin and abstract skeleton, a dessicated substructure, is simply the pragmatic nature of science, not any philosophical consideration. Bergson, Leroy, Poincaré, Meyerson, Brunschvicg and many others have, Evola writes,

“brought to light the altogether practical and pragmatic character of scientific methods. The more ‘comfortable’ ideas and theories become ‘true’, in regard to the organization of the data of sensorial experience. A choice between such data is made consciously or instinctively, excluding systematically those that do not lend themselves to being controlled; thus also everything qualitative and unrepeatable that is not susceptible to being mathematized….Scientific ‘objectivity’ consists solely in being ready at any moment to abandon existing theories of hypotheses, as soon as the chance appears for the better control of reality.”

Such science cannot yield as a result a philosophic conception of the substructure comparable to the Platonic forms even as conceived by Plato in his late, mathematical phase. Ward is right, for all I know, with regard to Penrose. But how representative is he?

Evola does argue that Einstein’s theory of relativity “has brought us…closer to absolute certainties”. “Only the profane”, he writes, “in hearing talk of relativity, could believe that the new theory had destroyed every certainty and almost sanctioned a kind of Pirandellian ‘thus it is, if you think so’.”  “A coherent syste of physics has been constructed to keep all relativity in check, to take every change and variation into account, with the greatest independence from points of reference and fro everything bound to observations, to the evidence of direct experience, and to current perceptions of space, time, and speed.” However, the system is, first of all, “of a purely formal character“. And second of all, it “is ‘absolute'” only “through the flexibility granted to it by its exclusively mathematical and algebraic nature”.

Its pure formality in itself disqualifies the system as ultimate reality, and its pragmatic use implies that there is no aspiration on its behalf to that status. “This theory”, Evola continues, “though far from common or philosophical relativism, is willing to admit the most unlikely relativities, but arms itself against them, so to speak, from the start. It intends to supply certainties that either leave out or anticipate them, and thus from the formal point of view are almost absolute. And if reality should ever revolt against them, a suitable readjustment of dimensions will restore these certainties.” Such a purely formal system, put to exclusively pragmatic use, cannot be ultimate reality; the Platonic forms cannot be reformulated in its terms.

This is roughly what Niels Bohr, one of the great founding fathers of quantum theory, thought. Bishop Berkeley was not so far wrong when he claimed that Locke’s “primary qualities” were in fact no more objectively real than the “secondary qualities” that were admitted to be mental constructs, or appearances to human forms of sensibility. Primary qualities are a sort of abstracted and idealised mathematical ground-plan of the rich sensory world of experience.

The primary qualities being reconceived in terms of “a sort of abstracted and idealised mathematical ground-plan”, they seem to be identified with the ultimate reality as here considered to be conceived by contemporary physicists. But if they are “no more objectively real” than the secondary qualities, if they too are “mental constructs, or appearances to human forms of sensibility”, their ultimacy is of course problematic, since the assertion that the mathematical ground-plan is the ultimate reality is a claim that it is precisely more objectively real than the world of secondary qualities.

Such a structure, which Ward here himself describes as “abstracted” and “idealised”, cannot, as such, explain “the rich sensory world of experience”. If the structure is abstracted, it must be abstracted from something, and that something is, from the perspective of our knowledge as finite beings, the world of primary qualities, the rich sensory world of experience. Ward therefore in this passage seems in reality to make the case for more than he explicitly intends. The fact that the mathematical ground-plan is, for us, abstracted, would seem to indicates that in itself, it is not ultimate in the sense of primary in relation to that which constitutes the ultimate ground of the experience of the secondary qualities. The very distinction between primary and secondary qualities collapses.

And again, the abstract, purely formal models of contemporary physics do not normally seem to be conceived of as an ultimately real ground-plan, but merely as a pragmatic tool. As I have already stressed, the pragmatic nature of modern science as such is what accounts for its the abstract formalism in the first place; Evola points out that there is nothing new in “the type of ‘certainty’ and knowledge to which Einstein’s theory leads’, that “his theory represents only the latest and most accessible manifestation of the characteristic orientation of all modern science”. Only it is taken to extraordinary extremes:

“The cosmic constant is a purely mathematical concept; in using it to speak of the speed of light, one no longer imagines speed, light, or propagation, one must only have in mind numbers and symbols. If someone were to ask those scientists what is light, without accepting an answer in mathematical symbols, they would look stupefied and not even understand the request. Everything that in recent physics proceeds from that stronghold participates rigorously in its nature: physics is completely algebraized. With the introduction of the concept of a ‘multidimensional continuum’ even that final sensible intuitive basis that survived in yesterday’s physics in the pure, schematic categories of geometrical space is reduced to mathematical formulae. Space and time here are one and the same; they form a ‘continuum’, itself expressed by algebraic functions. Together with the current, intuitive notion of space and time, that of force, energy, and movement also disappears…As in this algebraic scheme nothing remains of the concrete idea of force, even less so can there be room for cause.”

Evola is speaking here of some common-sense perceptions, but also to a considerable extent of one set of pragmatic concepts being replaced by another set. He does not consider – or have to consider in this context – the philosophical arguments with regard to the nature of what common sense perceives or of the first set of pragmatic concepts. The mathematical structure, being purely formal and pragmatically used, does not support the “‘spiritualization’ alleged by the popularizers…due to the disappearance of the idea of matter and the reduction of the concept of mass to that of energy”. This is “an absurdity, because mass and energy are made interchangeable values by an abstract formula. The only result of all this is a practical one: the application of the formula in order to control atomic forces. Apart from that, everything is consumed by the fire of algebraic abstraction associated with a radical experimentalism, that is, with a recording of simple phenomena.”

What Evola says here is that the Platonic interpretation is invalidated not just by the conception and use of the mathematical structure as a mere pragmatic instrument, but already by its abstract formality. Classical materialism’s idea of matter has indeed disappeared, and, as we have seen, there is more to the pragmatic concepts of contemporary physics than pure mathematics. But if the further, non-pragmatic, Platonic interpretation of the mathematization is to be legitimate, it would seem to follow from Evola’s analysis that something even more than the Penrosian interpretation would be required.

[Q]uantum physics seems to show that all that we really know of  [a real physical world in existence long before any human consciousness came into being] is how it appears to human consciousness, whether in perception or in mathematics or in some combination of both.

The “something” that, as Evola explained, is incomprehensible in itself and which science intends to subdue for practical ends – a “something” that is not matter as conceived by classical materialism – is still clearly in evidence in the last passage I cited from him (“atomic forces”, “simple phenomena”), alongside the pragmatically instrumentalized mathematical structure. Its being there of course means that this structure cannot in itself be the ultimate reality. “According to the most recent theory”, Evola writes, “purelymathematical entities that on the one hand magically spring forth in full irrationality, but on the other are ordered in a completely formal system of algebraic ‘production’, exhaustively account for everything that can be positively checked and formularized regarding the ultimate basis of sensible reality.” The ultimate basis is there, distinct from the (non-Platonic) “mathematical entities” which cannot really reach it, only “formularize”.

On Evola’s interpretation, with its particular evaluation of pragmatic science, we stand before the “definitive liquidation of all knowledge in the proper sense“, and Heisenberg “explicitly admitted this”: it is all “about a formal knowledge enclosed in itself, extremely precise in its practical consequences, in which, however, one cannot speak of knowledge of the real. For modern science, he says, ‘the object of research is no longer the object in itself, but nature as a function of the problems that man sets himself’; the logical conclusion in such science being that ‘henceforth man only meets himself’.”

As I remember it, Einstein too made statements to the same effect with regard to the relation between the formal knowledge and reality. But again, the latter, the “something” remains, on Evola’s own account, even as in the physics discussed by him man only meets himself. And Ward here returns to it in terms of “perception” alongside or in combination with mathematics. But he also makes the important idealistic point, not discussed by Evola, about their appearance in human consciousness which he finds confirmed by the recend developments in physics. Properly understood, what this implies is not that man only meets himself. It remains true in forms of idealism that go far beyond such limited, albeit at least methodologically collectivized (as it were) subjectivism and relativism.

And my general interpretation is confirmed. The perspective of human experience and knowledge does not provide any grounds for conceiving the mathematical, abstracted, at least originally, from rich, sensory perception, as alone representing ultimate reality and being a sufficient ground of it, of what Ward calls the “real”, “physical” world. If it is a “ground-plan” in some respects, as I certainly affirm but doubt that the majority of physicists affirm it must be, something must yet be added to it in order to account for concrete experience, and that something must be such that it is inadmissible to conceive of it as less of a ground than that part of the plan to which it is added. It seems it must, as I put it, be co-ultimate.

Ward’s argument points in the direction of the more complete idealistic case that he does not himself make. What we know of what Ward calls the “real”, “physical” world is certainly how it appears to human consciousness. Or rather, how that which appears appears to human consciousness, that which appears not in reality being the non-ultimately real, “physical” world, which is precisely the phenomenon-for-us alone, but ultimate reality itself. But that ultimate reality, conceived by us both through our reflection on perception (including the kind of inner perception that Wards does not consider here) and mathematics, must certainly be “more objectively real” than this phenomenal world; its very ultimacy is of course defined by its not being our “mental constructs” or “appearances to our forms of sensibility”. It cannot in itself be “abstracted and idealised”, but must be concretely ideal.

We apprehend what our human faculties of sense and mathematical creativity allow us to apprehend. And we have strong reason to think that things as they are in themselves do not correspond neatly to things as we apprehend them.

What we have “strong reason to think that tings as they are in themselves do not correspond neatly to” is matter as conceived by “old-fashioned classical materialism”. But I would stress that things being appearances to our finite consciousness (“mental constructs” has misleading connotations unless it is defined in terms of an adequately understood larger metaphysical position with regard to who does the constructing and how), does not entail that they are merely subjective and relative, although objectivity or gradually increased objectivity with regard to them, which is in reality an approximation to the absolute perspective, is an achievement of human thought in the process of knowledge.

“Things as we apprehend them” are real but not ultimately real. In one sense or on one level, they are in fact in reality as we apprehend them, once we have freed the apprehension from the illusion of common-sense materialism. The “rich sensory world” is not illusion, but real. But when we move beyond the Lockean meaning, “things as they are in themselves” can be understood to refer to their ideal ground, the ground of which they are, for us, an appearance. That ground, however, that ideal reality, must “correspond neatly” to the way we apprehend them inasmuch as it has to contain that which can account for the concrete richness that even the mere appearance possesses. Only thus do we reach the requisite completeness of our conception of the ideal ground-plan or ontic logos of reality. And that ground-plan or ontic logos must be conceived in strictly Platonic terms. Contemporary mathematical physicists who do accept that thereby of course support more of the case for idealism than the part of it that deals merely with contemporary physics’ abandonmet of classical materialism’s concept of matter.

But the ground-plan or ontic logos cannot be confused with the merely pragmatically used mathematical models of other such physicists. Evola ends his chapter on modern physics by clearly bringing out the difference between it’s position and that of what I summarize as the Platonic one:

“There is an aspect in which this latest natural science represents a type of inversion or counterfeit of that concept of catharsis, or purification, that in the traditional world was extended from the moral and ritual field to the intellectual; it referred to an intellectual discipline that, through overcoming the perceptions furnished by the animal senses and more or less mixed with the reactions of the I, would lead to a higher knowledge, to true knowledge. In effect, we have something similar in modern algebraized physics. Not only has it gradually freed itself from any immediate data of sense experience and common sense, buteven from all that which imagination could offer as support…Everything that can be suggested by the direct and living relationship of the observer to the observed is made unreal, irrelevant, and negligible. It is then like the catharsis that consumes every residue of the sensory, not in order to lead to a higher world, the ‘intelligible world’ or a ‘world of ideas’, as in the ancient schools of wisdom, but rather to the realm of pure mathematical thought, of number, of undifferentiated quantity, as opposed to the realm of quality, of meaningful forms and living forces: a spectral…world, an extreme intensification of the abstract intellect, where it is no longer a matter of things or phenomena, but almost of their shadows reduced to their common denominator, gray and indistinguishable. One may well speak of a falsification of the elevation of the mind above human sense-experience, which in the traditional world had as its effect not the destruction of the evidences of that experience, but their integration: the potentizing of the ordinary, concrete perception of natural phenomena by also experienceing their symbolic and intelligible aspects.”

The analysis of the basic character of contemporary physics here set forth is the basis of my questioning – for instance in an essay on Roger Kimball in Humanitas in 2001 –  of the accounts of some philosophers of science as representing the truth and the “deep” perspective on reality. We find the acceptance of that view of science also in philosophers whose work aims primarily at saving the world of of human experience from the effects of the scientific worldview, like Roger Scruton. There is certainly a deeper structure, but the one conceived of contemporary physics is not it. Both because of its pure formality and because of its pragmatic instrumentality, that mathematical structure simply does not represent a deeper truth than even ordinary human experience. This, on the other hand, does certainly not mean that it has no value. Nor does it mean that it cannot be reinterpreted and reformulated, as it perhaps is by Penrose, in strictly Platonic terms and thereby come to represent one side of a real theory of forms or ontic logos.

Mathematical physics does at present certainly support the part of the case against materialism that is Ward’s main point this far. On that point I agree with him. It has also been necessary to point out that mathematical physics does not in itself represent an adequate and complete idealist position. But that was of course not at all to be expected. Physics cannot take the place of metaphysics.

1 Response to “Mathematics and Reality”


  1. 1 Math James December 17, 2011 at 3:50 am

    There are just so many ways to learn math! Appreciate the post.


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